Section Summary: 13.5 (part a - lines in space)

1. Definitions

   The vector equation of a line L is

   

   where t is a scalar. Suppose that , , ; since this is true component-wise, we have that

   

   These are the parametric equations of the line L.

   These equations are not unique to a line: any point and vector with orientation along the line will give another set of equations.

   We can solve for t in each of the three parametric equations above to get a set of three equations

   

   called the symmetric equations of L.

   Skew lines are lines that do not intersect and are not parallel. Note: this can't happen in the plane! It only happens in three-space (or higher), that lines can pass like ships in the night....

2. Theorems

   None to speak of.

3. Properties/Tricks/Hints/Etc.

   Lines in 3-space (or higher) can pair up in only one of three ways:

   • parallel
   • intersecting
   • skew (missing each other completely yet not oriented in the same way)

4. Summary

   In the first part of section 13.5, we are introduced to various ways of thinking about lines in space. We meet with several different equations of space lines, and see that lines in space behave a little differently than lines in the plane.